Chapter 8 Well Models

7/31/2019 Chapter 8 Well Models

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Chapter 8 Well Models

Reading assignment: Chapter 7 in Mattax and Dalton, Reservoir Simulation

Up to now we have treated wells are point or line sources or sinks with

specified rates. In practice the well rates may be the quantity desired to bepredicted. The control variable that an operator has in producing a reservoir maybe the surface or well head pressure in the case of a flowing well or the bottomhole pressure in the case of a pumping well. There may be a hierarchy ofcontrols governing the production or injection between the following: (1) gridblocks, (2) sand members, (3) well, (4) group of wells, (5) reservoir, and (6) field.Here we will treat only the most fundamental level, i.e., the relation between thereservoir variables and well variables, i.e., bottom hole or surface pressure.

8.1 Relation with Grid Block

Rather than considering a well asa line source or sink, the well isconsidered as an interior boundarycondition with a radius, rw, flow rate, q,and pressure, pw. See Fig. 8.1a. Thefollowing assumptions are made.1) The flow from (to) the grid block to(from) the well is treated asaxisymmetric in an annular region withexternal radius, re, where

/er x y = , and internal radius, rw,where rw


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(i.e.,not function of r)

at , 2

implies :

2

2

r

w w r

r

k k b dpr constant

dr

r r r h b u q

qconst

hdp q

rdr h k k b

=

= =

=

=

(8.1b)

Let:

( )2

/

rD w

D w

h k k bp p p

q

r r r

=

=

(8.1c)

1

ln

ln 2

at 1, 0, 2 0

DD

D

DD D

D

D D

D D

dpr

dr

drdp d r

r

p r const

r p const

=

= =

= +

= = =

(8.1d)

This equation gives the pressure profile. The computed grid block pressurecorresponds to the average pressure in the annular region.

( )

/

1

/

1

/

1

2

2

2

1/

2

e w

e w

e w

r r

D D D

D r r

D D

r r

D D D

e w

r p dr p

r dr

r p dr

r r

=

(8.1e)

(2

ln 2 ln 1

4

rr r dr r =

) (8.1f)

8-2


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( ) ( )( )

( )

( )

( )

2

2

1/ 2 ln /

41

/2

ln / 1/ 2

2

e w e w

D

e w

e w

rw

r r r r

p

r r

r r

h k k bp p

q

=

=

1

(8.1g)

( )

( )( )

( )

( )( )

2

ln / 1/ 2

or in oil field units

2

887.2 ln / 1/ 2

wr

e w

wr

e w

p ph k k bq

r r

p ph k k bq

r r

=

=

(8.1h)

whereq(STB/d) b(STB/RB)

h(ft) (cp)k(md) p(psi)

The proceeding calculations were for a well in the center of a grid block. Ifa model is an element of symmetry, then the element may contain only a half orquarter well as illustrated in Fig. 8.1b.

x

yre

rw pw

x

yre

rwpw

Half Well Quarter Well Fig. 8.1 b Half and quarter well in a grid block

8-3


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In the cases of the half well the relation between the equivalent radius and gridblock dimensions are as follows.

2 2

2

e

e

r x y

x y

r

=

=

(8.1i)

In the cases of the quarter well the relation between the equivalent radius andgrid block dimensions are as follows.

2 4

4

e

e

r x y

x yr

=

=

(8.1j)

In the case of the half well and quarter well, the factor 2 in equation 8.1h is

replaced by and /2, respectively and the rates are one half or one quarter ofthe whole well. All these modifications for whole, half, or quarter well can beencompassed by defining a well fraction parameter, WFRAC.

( )

( )( )

/

2

ln / 1 / 2

e FRAC

wr FRAC

e w

r x y W

p ph k k b W q

r r

=

=

The expression for a well with cylindrical coordinates was given inequation 7.2g. The sign is changed to denote production as negative and

injection as positive rates.

( ) ( ) ( )

( )( ) ( )

1 1

1

1

, , ln2

2, ,

ln /

w w

w

w

q/p r t p r t r r

k h

k hq p r t p

r rr t

= +

=

Mattax and Dalton (1990) references several alternative well inflowcoefficients. Peaceman (1978) found that expression that best described steadyflow in a square domain with Cartesian coordinates is different from Eq. (8.1h). A

skin factor is introduced to find the equivalence between the two models.

8-4


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( )

( )( )

( )( )

( )

2,Eq(8.1h)

ln / 1 / 2

/ , whole well

2, Peaceman,ln 0.2 /

log 0.2 1 / 2

0.54

o wr

e w

e

o wr

w

p ph k k bq

r r s

r x y

p ph k k b

q yx r

s

=

+

=

= =

= +

=

x

The use of this pseudo-skin factor will match the analytical solution whenthe well is in the center of a square domain. However, the original expression(with WFRAC) matches the analytical solution for a quarter-well at the corner ofthe one-quarter element of symmetry.

Equation 8.1h is abbreviated by calling the coefficient the productivityindex, PI, (or injectivity index) and identifying the average pressure as the gridblock pressure, pij. The well rate is made semi-implicit in the grid block pressureand saturation.

( )

( ) ( ) ( )(

( )

)

( )

1 1

ij ij w

n n n n n

ij w ij ij ij w ij ij

n n

ij w t ij ij w t ij

q PI p p

dPIPI p p PI p p p p S S

dS

dPIPI p p PI p p p S

dS

+ +

=

=

=

n

)

)

n w

(8.1k)

The latter two terms called the "implicit coefficients" and are very important forstability. The importance of this semi-implicit formulation is illustrated by a simplestability analysis for the pressure equation. The well rate and accumulationterms in equation 6.6ll are all that are of interest for the stability analysis.

(8.1l)( )

( ) (

1

( )

1

n n

IMP

n w n n

C p p q

PI p p PI p p

+

+

=

=

The implicit coefficient, , is zero for the explicit rate formulation and is

equal to the PIfor the semi-implicit well formulation. The equation is rewritten asfollows.

(IMPPI

(8.1m)( ) ( )( ) ( )1IMP IMPnC PI p C PI PI p PI p++ = + +

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Suppose en is the difference between two solutions of the difference equationthat differ by a small amount at one time step. The difference equation for enafter this time step is as follows.

( ) (

( )( )

)( ) ( )1( )

1 ( )

IMP IMP

n n

IMP

n nIMP

C PI e C PI PI e

C PI PI e e

C PI

+

+

+ = +

+ =

+

(8.1n)

The coefficient of the second equation is an amplification factor. For the semi-implicit formulation, the amplification factor is always less than unity. With theexplicit formulation, the factor becomes negative for PI > Cand is negative with amagnitude greater than unity if PI > 2C. In the latter case, the system isunstable.

Assignment 8.1 Well model with rate and pressure constraints a) Simulatethe problem of assignment 7.1 with NX=10 and following well model. Theproductivity index for this cylindrical system is given by equation 7.2g. Let thewell radius and kh be as in assignment 7.1. Specify the minimum bottom holepressure to be 14.7 psi. b) Use the same pressure constraint for the one-quarterwell in the corner of a square as in assignment 6.4. c) Use the same pressureconstraint for a whole well in the center of a 2000 by 2000 square. Find thepseudo-skin factor that matches the analytical solution. In all cases compare thewell pressure with the analytical solution for constant rate.

8.2 Well Impairment

The previous section assumed that the formation properties werehomogeneous throughout the grid block. In practice, the formation usually doesnot have uniform properties next to the well. The formation may have beeninvaded by the drilling mud solids, precipitation of pressure sensitive mineralsmay have resulted is scaling, the formation may have compacted due to pressuredepletion, or the well may be cased and cemented with perforations providing thecommunication with the formation. The combined effect of the departure from ahomogeneous formation is represented by a "skin factor" that can be determinedby transient pressure analysis. It is assumed that the flow is axisymmetric andthe permeability distribution is described as in Figure 8.2a.

The same assumptions are made as before except that the formation is

8-6

r w

r s r e

k s H o m o g e n e o u s k

S k i n

Fig. 8.2a Schematic of well with skin of modified permeability, ks


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homogeneous except for a region of radius, rs, in which the permeability is ks.

( )1

D DD

D

k r dpr

k dr= (8.2a)

/ , 1 /

1, / /

s D s wDD

s w D e wD

k k r r r dpr

r r r r r dr

<


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The effect of the skin factor is be examined by defining an "effective well

bore radius", ' .wr

(8.2h)( ) ( )ln / ' ln / e w e wr r r r s= +

)

(

ln ' ln

ln

w w

s

w

r r s

r e

=

=(8.2i)

(8.2j)( )

( / 1

'

/s

s

w w

k k

w s w

r r e

r r r

=

=)

if , '

if , 'if , '

if , '

s w w

s w w

s w w

s w

k k r r

k k r r k k r r r

k r

<

Chapter 8 Well Models

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